Free energy in multi-species mixed $p$-spin spherical models
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866918085487755264 |
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| author | Bates, Erik Sohn, Youngtak |
| author_facet | Bates, Erik Sohn, Youngtak |
| contents | We prove a Parisi formula for the limiting free energy of multi-species spherical spin glasses with mixed $p$-spin interactions. The upper bound involves a Guerra-style interpolation and requires a convexity assumption on the model's covariance function. Meanwhile, the lower bound adapts the cavity method of Chen so that it can be combined with the synchronization technique of Panchenko; this part requires no convexity assumption. In order to guarantee that the resulting Parisi formula has a minimizer, we formalize the pairing of synchronization maps with overlap measures so that the constraint set is a compact metric space. This space is not related to the model's spherical structure and can be carried over to other multi-species settings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_14790 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Free energy in multi-species mixed $p$-spin spherical models Bates, Erik Sohn, Youngtak Probability Disordered Systems and Neural Networks Mathematical Physics 60K35, 60G15, 82B44, 82D30 We prove a Parisi formula for the limiting free energy of multi-species spherical spin glasses with mixed $p$-spin interactions. The upper bound involves a Guerra-style interpolation and requires a convexity assumption on the model's covariance function. Meanwhile, the lower bound adapts the cavity method of Chen so that it can be combined with the synchronization technique of Panchenko; this part requires no convexity assumption. In order to guarantee that the resulting Parisi formula has a minimizer, we formalize the pairing of synchronization maps with overlap measures so that the constraint set is a compact metric space. This space is not related to the model's spherical structure and can be carried over to other multi-species settings. |
| title | Free energy in multi-species mixed $p$-spin spherical models |
| topic | Probability Disordered Systems and Neural Networks Mathematical Physics 60K35, 60G15, 82B44, 82D30 |
| url | https://arxiv.org/abs/2109.14790 |